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Datei: chain_rule.prf   Sprache: Lisp

Original von: PVS©

(chain_rule
 (chain_rule_cnvg_TCC1 0
  (chain_rule_cnvg_TCC1-1 nil 3475835686
   ("" (lemma deriv_domain1)
    (("" (expand "deriv_domain?") (("" (propax) nil nil)) nil)) nil)
   ((deriv_domain1 formula-decl nil chain_rule nil)) nil))
 (chain_rule_cnvg_TCC2 0
  (chain_rule_cnvg_TCC2-1 nil 3475835686
   ("" (lemma not_one_element1)
    (("" (expand "not_one_element?") (("" (propax) nil nil)) nil)) nil)
   ((not_one_element1 formula-decl nil chain_rule nil)) nil))
 (chain_rule_cnvg_TCC3 0
  (chain_rule_cnvg_TCC3-1 nil 3475835686
   ("" (skosimp*)
    (("" (lemma deriv_domain2)
      (("" (expand "deriv_domain?") (("" (inst?) nil nil)) nil)) nil))
    nil)
   ((deriv_domain2 formula-decl nil chain_rule nil)) nil))
 (chain_rule_cnvg_TCC4 0
  (chain_rule_cnvg_TCC4-1 nil 3475835686
   ("" (lemma not_one_element2)
    (("" (skosimp*)
      (("" (expand "not_one_element?") (("" (inst?) nil nil)) nil))
      nil))
    nil)
   ((not_one_element2 formula-decl nil chain_rule nil)) nil))
 (chain_rule_cnvg 0
  (chain_rule_cnvg-1 nil 3475835691
   ("" (skosimp)
    ((""
      (auto-rewrite "deriv_TCC[T1]" "deriv_TCC[T2]"
                    "adherence_fullset[T1]" "adherence_fullset[T2]")
      (("1" (use "derivative_equivalence1[T1]")
        (("1" (replace -2)
          (("1" (prop)
            (("1" (forward-chain "derivable_continuous[T1]")
              (("1" (delete -2 -3)
                (("1" (rewrite continuity_def)
                  (("1" (rewrite "derivative_equivalence2[T1]")
                    (("1" (rewrite "derivative_equivalence2[T1]")
                      (("1" (rewrite "derivative_equivalence2[T2]")
                        (("1" (skosimp*)
                          (("1"
                            (inst 1
                             "DG!1 * phi!1 + DF!1 * (phi!2 o f!1) + phi!1 * (phi!2 o f!1)")
                            (("1" (prop)
                              (("1"
                                (use
                                 "convergence_composition"
                                 ("f" "f!1" "g" "phi!2" "x" "x!1"))
                                (("1"
                                  (assert)
                                  (("1"
                                    (delete -4 -5 -6)
                                    (("1"
                                      (lemma "cv_sum[T1]")
                                      (("1"
                                        (inst
                                         -
                                         "x!1"
                                         "DG!1 * phi!1"
                                         "DF!1 * (phi!2 o f!1) + phi!1 * (phi!2 o f!1)"
                                         "0"
                                         "0")
                                        (("1"
                                          (assert)
                                          (("1"
                                            (split -1)
                                            (("1"
                                              (expand "+")
                                              (("1" (propax) nil nil))
                                              nil)
                                             ("2"
                                              (hide -1 -2 2)
                                              (("2"
                                                (lemma "cv_scal[T1]")
                                                (("2"
                                                  (inst
                                                   -
                                                   "x!1"
                                                   "DG!1"
                                                   "phi!1"
                                                   "0")
                                                  (("2"
                                                    (assert)
                                                    nil
                                                    nil))
                                                  nil))
                                                nil))
                                              nil)
                                             ("3"
                                              (hide 2)
                                              (("3"
                                                (lemma "cv_sum[T1]")
                                                (("3"
                                                  (inst
                                                   -
                                                   "x!1"
                                                   "DF!1 * (phi!2 o f!1)"
                                                   "phi!1 * (phi!2 o f!1)"
                                                   "0"
                                                   "0")
                                                  (("3"
                                                    (assert)
                                                    (("3"
                                                      (hide 2)
                                                      (("3"
                                                        (lemma
                                                         "cv_scal[T1]")
                                                        (("3"
                                                          (inst
                                                           -
                                                           "x!1"
                                                           "DF!1"
                                                           "(phi!2 o f!1)"
                                                           "0")
                                                          (("3"
                                                            (assert)
                                                            (("3"
                                                              (lemma
                                                               "cv_prod[T1]")
                                                              (("3"
                                                                (inst
                                                                 -
                                                                 "x!1"
                                                                 "phi!1"
                                                                 "(phi!2 o f!1)"
                                                                 "0"
                                                                 "0")
                                                                (("3"
                                                                  (assert)
                                                                  nil
                                                                  nil))
                                                                nil))
                                                              nil))
                                                            nil))
                                                          nil))
                                                        nil))
                                                      nil))
                                                    nil))
                                                  nil))
                                                nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2"
                                (delete -1 -2 -4)
                                (("2"
                                  (skolem!)
                                  (("2"
                                    (inst -1 "y!1")
                                    (("2"
                                      (inst -2 "f!1(y!1)")
                                      (("2"
                                        (expand "o ")
                                        (("2"
                                          (expand "*")
                                          (("2"
                                            (expand "+ ")
                                            (("2"
                                              (replace -2)
                                              (("2" (assertnil nil))
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil))
                          nil)
                         ("2" (lemma "deriv_domain2")
                          (("2" (lemma "not_one_element2")
                            (("2" (expand "not_one_element?")
                              (("2" (propax) nil nil)) nil))
                            nil))
                          nil))
                        nil)
                       ("2" (lemma "not_one_element1")
                        (("2" (expand "not_one_element?")
                          (("2" (propax) nil nil)) nil))
                        nil)
                       ("3" (lemma "deriv_domain1 ")
                        (("3" (propax) nil nil)) nil))
                      nil)
                     ("2" (lemma "not_one_element1")
                      (("2" (expand "not_one_element?")
                        (("2" (propax) nil nil)) nil))
                      nil)
                     ("3" (lemma "deriv_domain1 ")
                      (("3" (propax) nil nil)) nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil)
         ("2" (lemma "not_one_element1")
          (("2" (expand "not_one_element?") (("2" (propax) nil nil))
            nil))
          nil)
         ("3" (lemma "deriv_domain1 ") (("3" (propax) nil nil)) nil))
        nil)
       ("2" (lemma "not_one_element2") (("2" (propax) nil nil)) nil)
       ("3" (lemma "deriv_domain2 ") (("3" (propax) nil nil)) nil))
      nil))
    nil)
   ((deriv_domain? const-decl "bool" deriv_domain_def nil)
    (not_one_element? const-decl "bool" deriv_domain_def nil)
    (T2 formal-subtype-decl nil chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" chain_rule nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil)
    (derivable_continuous formula-decl nil derivatives_def nil)
    (continuity_def formula-decl nil continuous_functions nil)
    (O const-decl "T3" function_props nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (cv_sum formula-decl nil lim_of_functions nil)
    (cv_prod formula-decl nil lim_of_functions nil)
    (cv_scal formula-decl nil lim_of_functions nil)
    (convergence_composition formula-decl nil lim_of_composition nil)
    (+ const-decl "[T -> real]" real_fun_ops "reals/")
    (* const-decl "[T -> real]" real_fun_ops "reals/")
    (* const-decl "[T -> real]" real_fun_ops "reals/")
    (not_one_element1 formula-decl nil chain_rule nil)
    (deriv_domain1 formula-decl nil chain_rule nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (real_plus_real_is_real application-judgement "real" reals nil)
    (derivative_equivalence2 formula-decl nil derivatives_alt nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (T1 formal-subtype-decl nil chain_rule nil)
    (T1_pred const-decl "[real -> boolean]" chain_rule nil)
    (derivative_equivalence1 formula-decl nil derivatives_alt nil)
    (not_one_element2 formula-decl nil chain_rule nil)
    (deriv_domain2 formula-decl nil chain_rule nil))
   nil))
 (composition_derivable_TCC1 0
  (composition_derivable_TCC1-1 nil 3313840193
   ("" (skosimp)
    (("" (lemma "deriv_domain2")
      (("" (expand "deriv_domain?") (("" (propax) nil nil)) nil)) nil))
    nil)
   ((deriv_domain2 formula-decl nil chain_rule nil)) nil))
 (composition_derivable_TCC2 0
  (composition_derivable_TCC2-1 nil 3313840193
   ("" (skosimp)
    (("" (lemma "not_one_element2")
      (("" (expand "not_one_element?") (("" (propax) nil nil)) nil))
      nil))
    nil)
   ((not_one_element2 formula-decl nil chain_rule nil)) nil))
 (composition_derivable 0
  (composition_derivable-3 nil 3475836001
   ("" (expand "derivable?")
    (("" (expand "convergent?")
      (("" (skosimp*)
        (("" (forward-chain "chain_rule_cnvg") (("" (inst?) nil nil))
          nil))
        nil))
      nil))
    nil)
   ((convergent? const-decl "bool" lim_of_functions nil)
    (chain_rule_cnvg formula-decl nil chain_rule nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T1_pred const-decl "[real -> boolean]" chain_rule nil)
    (T1 formal-subtype-decl nil chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" chain_rule nil)
    (T2 formal-subtype-decl nil chain_rule nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (derivable? const-decl "bool" derivatives_def nil))
   nil)
  (composition_derivable-2 nil 3445340288
   ("" (expand "derivable?")
    (("" (expand "convergent?")
      (("" (skosimp*)
        (("" (forward-chain "chain_rule") (("" (inst?) nil nil)) nil))
        nil))
      nil))
    nil)
   ((convergent? const-decl "bool" lim_of_functions nil)
    (derivable? const-decl "bool" derivatives_def nil))
   nil)
  (composition_derivable-1 nil 3313840193
   ("" (expand "derivable")
    (("" (expand "convergent?")
      (("" (skosimp*)
        (("" (forward-chain "chain_rule") (("" (inst?) nil nil)) nil))
        nil))
      nil))
    nil)
   ((convergent? const-decl "bool" lim_of_functions nil)) nil))
 (composition_derivable_fun_TCC1 0
  (composition_derivable_fun_TCC1-1 nil 3313840193
   ("" (skosimp*)
    (("" (lemma "deriv_domain2") (("" (propax) nil nil)) nil)) nil)
   ((deriv_domain2 formula-decl nil chain_rule nil)) nil))
 (composition_derivable_fun_TCC2 0
  (composition_derivable_fun_TCC2-1 nil 3313840193
   ("" (skosimp*)
    (("" (lemma not_one_element2) (("" (propax) nil nil)) nil)) nil)
   ((not_one_element2 formula-decl nil chain_rule nil)) nil))
 (composition_derivable_fun 0
  (composition_derivable_fun-1 nil 3313840193
   ("" (expand "derivable?")
    (("" (skosimp*)
      (("" (rewrite "composition_derivable")
        (("1" (inst?) nil nil) ("2" (inst? -2) nil nil)) nil))
      nil))
    nil)
   ((T2 formal-subtype-decl nil chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" chain_rule nil)
    (T1 formal-subtype-decl nil chain_rule nil)
    (T1_pred const-decl "[real -> boolean]" chain_rule nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (composition_derivable formula-decl nil chain_rule nil)
    (derivable? const-decl "bool" derivatives nil))
   nil))
 (deriv_composition_TCC1 0
  (deriv_composition_TCC1-1 nil 3313840193
   ("" (lemma "composition_derivable")
    (("" (skosimp*)
      (("" (inst?) (("" (expand "O") (("" (assertnil nil)) nil))
        nil))
      nil))
    nil)
   ((O const-decl "T3" function_props nil)
    (T2 formal-subtype-decl nil chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" chain_rule nil)
    (T1 formal-subtype-decl nil chain_rule nil)
    (T1_pred const-decl "[real -> boolean]" chain_rule nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (composition_derivable formula-decl nil chain_rule nil))
   nil))
 (deriv_composition 0
  (deriv_composition-1 nil 3313840193
   ("" (skosimp)
    (("" (forward-chain "composition_derivable")
      ((""
        (auto-rewrite "deriv_TCC[T1]" "deriv_TCC[T2]"
                      ("deriv[T1]" "derivable?[T1]" "deriv[T2]"
                       "derivable?[T2]")
                      ("lim_fun_lemma[(A[T1](x!1))]"
                       "lim_fun_lemma[(A[T2](f!1(x!1)))]"))
        (("1" (assert)
          (("1" (rewrite "lim_fun_def[(A[T1](x!1))]")
            (("1" (rewrite "chain_rule_cnvg"nil nil)) nil))
          nil)
         ("2" (skosimp*)
          (("2" (lemma "not_one_element2")
            (("2" (expand "not_one_element?") (("2" (inst?) nil nil))
              nil))
            nil))
          nil)
         ("3" (lemma "deriv_domain2 ")
          (("3" (skosimp*)
            (("3" (expand "deriv_domain?") (("3" (inst?) nil nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((composition_derivable formula-decl nil chain_rule nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T1_pred const-decl "[real -> boolean]" chain_rule nil)
    (T1 formal-subtype-decl nil chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" chain_rule nil)
    (T2 formal-subtype-decl nil chain_rule nil)
    (deriv_domain2 formula-decl nil chain_rule nil)
    (not_one_element2 formula-decl nil chain_rule nil)
    (real_times_real_is_real application-judgement "real" reals nil)
    (deriv const-decl "real" derivatives_def nil)
    (derivable? const-decl "bool" derivatives_def nil)
    (lim_fun_lemma formula-decl nil lim_of_functions nil)
    (chain_rule_cnvg formula-decl nil chain_rule nil)
    (A const-decl "setof[nzreal]" derivatives_def nil)
    (setof type-eq-decl nil defined_types nil)
    (nzreal nonempty-type-eq-decl nil reals nil)
    (/= const-decl "boolean" notequal nil)
    (lim const-decl "{l: real | convergence(f, x0, l)}"
     lim_of_functions nil)
    (convergence const-decl "bool" lim_of_functions nil)
    (convergent? const-decl "bool" lim_of_functions nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (O const-decl "T3" function_props nil)
    (NQ const-decl "real" derivatives_def nil)
    (lim_fun_def formula-decl nil lim_of_functions nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (not_one_element? const-decl "bool" deriv_domain_def nil)
    (deriv_domain? const-decl "bool" deriv_domain_def nil))
   nil))
 (gg_TCC1 0
  (gg_TCC1-1 nil 3313840193
   ("" (lemma "deriv_domain2") (("" (propax) nil nil)) nil)
   ((deriv_domain2 formula-decl nil chain_rule nil)) nil))
 (gg_TCC2 0
  (gg_TCC2-1 nil 3313840193
   ("" (lemma "not_one_element2") (("" (propax) nil nil)) nil)
   ((not_one_element2 formula-decl nil chain_rule nil)) nil))
 (deriv_comp_fun_TCC1 0
  (deriv_comp_fun_TCC1-1 nil 3313840193
   ("" (skosimp) (("" (rewrite "composition_derivable_fun"nil nil))
    nil)
   ((composition_derivable_fun formula-decl nil chain_rule nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T1_pred const-decl "[real -> boolean]" chain_rule nil)
    (T1 formal-subtype-decl nil chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" chain_rule nil)
    (T2 formal-subtype-decl nil chain_rule nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (derivable? const-decl "bool" derivatives nil))
   nil))
 (deriv_comp_fun 0
  (deriv_comp_fun-1 nil 3313840193
   ("" (skolem-typepred)
    (("" (assert)
      (("" (expand "derivable?")
        (("" (auto-rewrite "composition_derivable_fun")
          (("" (apply-extensionality :hide? t)
            (("" (expand "*")
              (("" (expand "o" 1 2)
                (("" (expand "deriv")
                  (("" (rewrite "deriv_composition")
                    (("1" (inst? -1) nil nil) ("2" (inst? -2) nil nil))
                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((real_times_real_is_real application-judgement "real" reals nil)
    (deriv_composition formula-decl nil chain_rule nil)
    (deriv_fun type-eq-decl nil derivatives nil)
    (deriv const-decl "[T -> real]" derivatives nil)
    (O const-decl "T3" function_props nil)
    (* const-decl "[T -> real]" real_fun_ops "reals/")
    (T2 formal-subtype-decl nil chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" chain_rule nil)
    (derivable? const-decl "bool" derivatives nil)
    (T1 formal-subtype-decl nil chain_rule nil)
    (T1_pred const-decl "[real -> boolean]" chain_rule nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number nonempty-type-decl nil numbers nil)
    (NOT const-decl "[bool -> bool]" booleans nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (boolean nonempty-type-decl nil booleans nil))
   nil))
 (comp_derivable_fun 0
  (comp_derivable_fun-1 nil 3475834576
   ("" (skosimp*)
    (("" (lemma "composition_derivable_fun")
      (("" (expand "o ") (("" (inst?) (("" (assertnil nil)) nil))
        nil))
      nil))
    nil)
   ((composition_derivable_fun formula-decl nil chain_rule nil)
    (number nonempty-type-decl nil numbers nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (real nonempty-type-from-decl nil reals nil)
    (T1_pred const-decl "[real -> boolean]" chain_rule nil)
    (T1 formal-subtype-decl nil chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" chain_rule nil)
    (T2 formal-subtype-decl nil chain_rule nil)
    (O const-decl "T3" function_props nil))
   shostak))
 (chain_rule_TCC1 0
  (chain_rule_TCC1-1 nil 3475835726
   ("" (skosimp*)
    (("" (lemma comp_derivable_fun)
      (("" (inst?) (("" (assertnil nil)) nil)) nil))
    nil)
   ((comp_derivable_fun formula-decl nil chain_rule nil)
    (derivable? const-decl "bool" derivatives nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (T2 formal-subtype-decl nil chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" chain_rule nil)
    (T1 formal-subtype-decl nil chain_rule nil)
    (T1_pred const-decl "[real -> boolean]" chain_rule nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil))
   nil))
 (chain_rule 0
  (chain_rule-1 nil 3475853574
   ("" (skosimp*)
    (("" (lemma deriv_comp_fun)
      (("" (expand "o ") (("" (inst?) nil nil)) nil)) nil))
    nil)
   ((deriv_comp_fun formula-decl nil chain_rule nil)
    (derivable? const-decl "bool" derivatives nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (T2 formal-subtype-decl nil chain_rule nil)
    (T2_pred const-decl "[real -> boolean]" chain_rule nil)
    (T1 formal-subtype-decl nil chain_rule nil)
    (T1_pred const-decl "[real -> boolean]" chain_rule nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (O const-decl "T3" function_props nil))
   shostak)))


¤ Dauer der Verarbeitung: 0.8 Sekunden  (vorverarbeitet)  ¤





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